University of Chicago - Geometric Analysis Seminar

Department of Mathematics

Wee welcome all those who are interested to join us on Tuesdays, 3:00–4:00 PM, Ryerson Laboratory, Room 358, unless noted otherwise.

Everyone is also invited to attend the Learning Seminar on Geometric Analysis, organized by Prof. Andre Neves. More information on http://www.math.uchicago.edu/~geometric_analysis/learning


SPRING 2017


Apr 4, 2017 Tue, 3:00–4:00PM, Eckhart 358

Shmuel Weinberger, The University of Chicago

Persistent Homology and Gromov's theorem on closed contractible geodesics. : Gromov showed that on any closed Riemannian manifold whose fundamental group has unsolvable word problem, there are infinitely many closed nullhomotopic geodesics (of index 0). I will explain this theorem from the point of Persistent Homology of the free loop space, and then give some refinements (e.g. to less exotic fundamental groups) and extensions (to some other functionals other than energy on loops).


Apr 11, 2017 Tue, 3:00–4:00PM, Eckhart 358

David Wiygul, UC Irvine

The Bartnik-Bray outer mass of small spheres : In 1989 Robert Bartnik proposed a definition of quasilocal mass in general relativity. The Bartnik mass is known to enjoy several attractive properties but is not straightforward to evaluate. I will talk about a first-order estimate for a natural modification of Bartnik's definition applied to small perturbations of spheres in Euclidean space. In particular I will describe an application to the small-sphere limit in time-symmetric slices.


Apr 18, 2017 Tue, 3:00–4:00PM, Eckhart 358

Nick Edelen , MIT

Quantitative Reifenberg for Measures : In joint work with Aaron Naber and Daniele Valtorta, we demonstrate a quantitative structure theorem for measures in R^n under assumptions on the Jones \beta-numbers, which measure how close the support is to being contained in a subspace. Measures with this property have arisen in several interesting scenarios: in obtaining packing estimates on and rectifiability of the singular set of minimal surfaces; in characterizing L2-boundedness of Calderon-Zygmund operators; and as an "annalist's" formulation of the travelling salesman problem.


Apr 25, 2017 Tue, 3:00–4:00PM, Eckhart 358

Dan Ketover , Princeton University

Free boundary minimal surfaces of unbounded genus : Free boundary minimal surfaces are natural variational objects that have been studied since the 40s. In spite of this, very few explicit examples in the simplest case of the round three ball are known. I will describe how variational methods can be used to construct new examples with unbounded genus resembling a desingularization of the critical catenoid and flat disk. I will also give a new variational interpretation of the previously known examples.


May 02, 2017 Tue, 3:00–4:00PM, Eckhart 358

Marco Radeschi , University of Notre Dame

Minimal hypersurfaces in compact symmetric spaces : A conjecture of Marques-Neves-Schoen says that for every embedded minimal hypersurface M in a manifold of positive Ricci curvature, the first Betti number of M is bounded above linearly by the index of M. We will show that for every compact symmetric space this result holds, up to replacing the index of M with its extended index. Moreover, for special symmetric spaces, the actual conjecture holds for all metrics in a neighbourhood of the canonical one. These results are a joint work with R. Mendes.


May 16, 2017 Tue, 3:00–4:00PM, Eckhart 358

Pierre Albin , University of Illinois at Urbana-Champaign

Analytic torsion of manifolds with fibered cusps : Analytic torsion is a spectral invariant of the Hodge Laplacian of a manifold with a flat connection. On a closed manifold it is equal to a topological invariant known as Reidemeister torsion. I will describe joint work with Frédéric Rochon and David Sher establishing a topological expression for the analytic torsion of a manifold with fibered cusp ends (such as a locally symmetric space of rank one). We establish our result by controlling the behavior of the spectrum along a degenerating class of Riemannian metrics.


May 23, 2017 Tue, 3:00–4:00PM, Eckhart 358 There will be two seminars on this date

Jacob Bernstein , Johns Hopkins University

Surfaces of Low Entropy : Following Colding and Minicozzi, we consider the entropy of (hyper)-surfaces in Euclidean space. This is a numerical measure of the geometric complexity of the surface and is intimately tied to to the singularity formation of the mean curvature flow. In this talk, I will discuss several results that show that closed surfaces for which the entropy is small are simple in various senses. This is all joint work with L. Wang.


May 23, 2017 Tue, 4:30–5:30PM, Eckhart 358 There will be two seminars on this date

Pedro Gaspar , IMPA

Minimal hypersurfaces and the Allen-Cahn equation on closed manifolds : Since the late 70s parallels between the theory of phase transitions and critical points of the area functional have helped us to understand variational properties of certain semi-linear elliptic PDEs and spaces of hypersurfaces which minimize the area in an appropriate sense. We will discuss some recent developments in this direction which extend well-known analogies regarding minimizers to more general variational solutions. In particular, borrowing ideas from the min-max theory of minimal hypersurfaces, we study the number of solutions of the Allen-Cahn equation in a closed manifolds and solutions with least non-trivial energy.


May 30, 2017 Tue, 3:00–4:00PM, Eckhart 358

Or Hershkovits , Stanford University

Uniqueness of mean curvature flow with mean convex singularities : Given a smooth compact hypersurface in Euclidean space, one can show that there exists a unique smooth evolution starting from it, existing for some maximal time. But what happens after the flow becomes singular? There are several notions through which one can describe weak evolutions past singularities, with various relationship between them. One such notion is that of the level set flow. While the level set flow is almost by definition unique, it has an undesirable phenomenon called fattening: Our "weak evolution" of n-dimensional hypersurfaces may develop (and does develop in some cases) an interior in R^{n+1}. This fattening is, in many ways, the right notion of non-uniqueness for weak mean curvature. As was alluded to above, fattening can not occur as long as the flow is smooth. Thus it is reasonable to say that the source of fattening is singularities. Permitting singularities, it is very easy to show that fattening does not occur if the initial hypersurface, and thus all the evolved hypersurface, are mean convex. Thus, singularities encountered during mean convex mean curvature flow should be of the kind that does not create singularities (i.e, the local structure of the singularities should prevent fattening, without any global mean convexity assumption). To put differently, it is reasonable to conjecture that: "An evolving surface cannot fatten unless it has a singularity with no spacetime neighborhood in which the surface is mean convex". In this talk, we will phrase a concrete formulation of this conjecture, and describe its proof. This is a joint work with Brian White.


June 05, 2017 Tue, 3:00–4:00PM, Eckhart 358

Xin Zhou , MIT

Min-max minimal hypersurfaces with free boundary : I will present a joint work with Martin Li. Minimal surfaces with free boundary are natural critical points of the area functional in compact smooth manifolds with boundary. In this talk, I will describe a general existence theory for minimal surfaces with free boundary. In particular, I will show the existence of a smooth embedded minimal hypersurface with free boundary in any compact smooth Euclidean domain. The minimal surfaces with free boundary were constructed using the min-max method. Our result allows the min-max free boundary minimal hypersurface to be improper; nonetheless the hypersurface is still regular.


June 29, 2017 Thu, 3:00–4:00PM, Eckhart 202

Henrik Matthiesen , Max Plank Institute for Mathematics Bonn

Existence of metrics maximizing the first eigenvalue on closed surfaces : We show that on each closed surface of fixed topological type, orientable or non-orientable, there is a metric, smooth away from finitely many conical singularities, that maximizes the first eigenvalue among all unit area metrics. The key new ingredient are several monotonicity results relating the corresponding maximal eigenvalues. This is joint work with Anna Siffert.


WINTER 2017


Jan 10, 2017 Tue, 3:00–4:00PM, Eckhart 312

Gang Liu, Northwestern University

On some recent progress of Yau's uniformization conjecture : Yau's uniformization conjecture states that a complete noncompact Kahler manifold with positive bisectional curvature is biholomorphic to the complex Euclidean space. We shall discuss some recent progress via the Gromov-Hausdorff convergence technique.


Jan 17, 2017 Tue, 3:00–4:00PM, Eckhart 312

John Ma, University of British Columbia

Compactness, finiteness properties of Lagrangian self-shrinkers in R^4 and Piecewise mean curvature flow. : In this talk, we discuss a compactness result on the space of compact Lagrangian self-shrinkers in R^4. When the area is bounded above uniformly, we prove that the entropy for the Lagrangian self-shrinking tori can only take finitely many values; this is done by deriving a Lojasiewicz-Simon type gradient inequality for the branched conformal self-shrinking tori. Using the finiteness of entropy values, we construct a piecewise Lagrangian mean curvature flow for Lagrangian immersed tori in R^4, along which the Lagrangian condition is preserved, area is decreasing, and the type I singularities that are compact with a fixed area upper bound can be perturbed away in finite steps. This is a Lagrangian version of the construction for embedded surfaces in R^3 by Colding and Minicozzi. This is a joint work with Jingyi Chen.


Jan 24, 2017 Tue, 3:00–4:00PM, Ryerson 358

Laura Schaposnik, University of Illinois at Chicago

On the geometry of branes in the moduli space of Higgs bundles. : After giving a gentle introduction to Higgs bundles, their moduli space and the associated Hitchin fibration, I shall describe how actions both on groups and on surfaces (anti-holomorphic or of finite groups) lead to families of interesting subspaces of the moduli space of Higgs bundles (the so-called branes). Finally, we shall look at correspondences between these branes that arise from Langlands duality, as well as from other relations between Lie groups.


Jan 31, 2017 Tue, 3:00–4:00PM, Ryerson 358

Pei-Ken Hung, Columbia University

A Minkowski inequality for hypersurfaces in the Anti-deSitter-Schwarzschild manifold : We prove a sharp inequality for hypersurfaces in the Anti-deSitter-Schwarzschild manifold. This inequality generalizes the classical Minkowski inequality for surfaces in the three dimensional Euclidean space, and has a natural interpretation in terms of the Penrose inequality for collapsing null shells of dust. The proof relies on a monotonicity formula for inverse mean curvature flow, and uses a geometric inequality established by Brendle.


Feb 7, 2017 Tue, 3:00–4:00PM, Ryerson 358

Lan-Hsuan Huang, University of Connecticut

Geometry of Static Asymptotically Flat 3-Manifolds : A static potential on a 3-manifold reflects the symmetry of the spacetime development and is also closely related to the scalar curvature deformation. We discuss how minimal surfaces of 3-manifold interact with the static potential and, as an application, give a new proof to the rigidity of the Riemannian positive mass theorem.


Feb 14, 2017 Tue, 3:00–4:00PM, Ryerson 358

Yu Li, University of Wisconsin-Madison

Ricci flow on asymptotically Euclidean manifolds : In this talk, we prove that if an asymptotically Euclidean (AE) manifold with nonnegative scalar curvature has long time existence of Ricci flow, it converges to the Euclidean space in the strong sense. By convergence, the mass will drop to zero as time tends to infinity. Moreover, in three dimensional case, we use Ricci flow with surgery to give an independent proof of positive mass theorem.


Feb 21, 2017 Tue, 3:00–4:00PM, Ryerson 358

Yannick Sire, Johns Hopkins University

Bounds on eigenvalues on Riemannian surfaces : In a joint program with N. Nadirashvili, we developed some tools to prove existence of extremal metrics in conformal classes for eigenvalues of the Laplace-Beltrami operator on surfaces. I will describe these results and move to an application to the isoperimetric inequality of the third eigenvalue on the 2-sphere.


Feb 28, 2017 Tue, 3:00–4:00PM, Ryerson 358 There will be two seminars on this date

Henri Roesch, Duke University

Proof of a Null Penrose conjecture using a new Quasi-local Mass : We define an explicit quasi-local mass functional which is non-decreasing along all foliations (satisfying a convexity assumption) of null cones. We use this new functional to prove the null Penrose conjecture under fairly generic conditions.


Feb 28, 2017 Tue, 4:00–5:00PM, Eckhart 206 There will be two seminars on this date - Note the different room

Luca Spolaor, MIT

A direct approach to an epiperimetric inequality for Free-Boundary problems : Using a direct approach, we prove a 2-dimensional epiperimetric inequality for the one-phase problem in the scalar and vectorial cases and for the double-phase problem. From this we deduce the $C^{1,\alpha}$ regularity of the free-boundary in the scalar one-phase and double-phase problems, and of the reduced free-boundary in the vectorial case, without any restriction on the sign of the component functions. In this talk I will try to explain the proof of the epiperimetric inequality in the scalar one-phase problem. This is joint work with Bozhidar Velichkov.


March 7, 2017 Tue, 3:00–4:00PM, Ryerson 358

Victoria Sadovskaya, Department of Mathematics - Penn State

Boundedness, compactness, and invariant norms for operator-valued cocycles.: We consider group-valued cocycles over dynamical systems with hyperbolic behavior, such as hyperbolic diffeomorphisms or subshifts of finite type. The cocycle A takes values in the group of invertible bounded linear operators on a Banach space and is Holder continuous. We consider the periodic data of A, i.e. the set of its return values along the periodic orbits in the base. We show that if the periodic data of A is bounded or contained in a compact set, then so is the cocycle. Moreover, in the latter case the cocycle is isometric with respect to a Holder continuous family of norms. This is joint work with B. Kalinin.


Mar 14, 2017 Tue, 3:00–4:00PM, Ryerson 358

Baris Coskunuser , Boston College

Embeddedness of the solutions to the H-Plateau Problem : In this talk, we will give a generalization of Meeks and Yau's embeddedness result on the solutions of the Plateau problem to the constant mean curvature disks. arXiv:1504.00661


FALL 2016


Oct 11, 2016 Tue, 3:00–4:00PM, Eckhart 207

Lu Wang, Department of Mathematics - University of Wisconsin-Madison

Asymptotic structure of self-shrinkers: We show that each end of a noncompact self-shrinker in Euclidean 3-space of finite topology is smoothly asymptotic to a regular cone or a self-shrinking round cylinder.


Oct 18, 2016 Tue, 3:00–4:00PM, Eckhart 207

Spencer Becker-Kahn, Department of Mathematics - MIT

Zero Sets of Smooth Functions and p-Sweepouts: It is well known that any closed set can occur as the zero set of a smooth function. But in many contexts in analysis, one does not work with arbitrary smooth functions; one works with smooth functions that vanish to only finite order. And the zero set of any such function must have some regularity and structure. With recourse to an analogy made by Gromov and some recent conjectures of Marques and Neves, I will explain the application of some general results about such zero sets to p-sweepouts (which are p-dimensional families of generalized hypersurfaces that sweepout a smooth manifold in a certain sense). In the course of doing so, I will explain a new result about the regularity of zero sets of smooth functions near a point of finite vanishing order (joint work with Tom Beck and Boris Hanin)..


Nov 8, 2016 Tue, 3:00–4:00PM, Eckhart 207 There will be two seminars on this date

Jonathan Zhu, Department of Mathematics - Harvard University

Entropy and self-shrinkers of the mean curvature flow: The Colding-Minicozzi entropy is an important tool for understanding the mean curvature flow (MCF), and is a measure of the complexity of a submanifold. Together with Ilmanen and White, they conjectured that the round sphere minimises entropy amongst all closed hypersurfaces. We will review the basics of MCF and their theory of generic MCF, then describe the resolution of the above conjecture, due to J. Bernstein and L. Wang for dimensions up to six and recently claimed by the speaker for all remaining dimensions. A key ingredient in the latter is the classification of entropy-stable self-shrinkers that may have a small singular set.


Nov 8, 2016 Tue, 4:00–5:00PM, Eckhart 207 There will be two seminars on this date

Greg Chambers, Department of Mathematics - The University of Chicago

Existence of minimal hypersurfaces in non-compact manifolds: I prove that every complete non-compact manifold of finite volume admits a minimal hypersurface of finite area. This work is in collaboration with Yevgeny Liokumovich.


Nov 15, 2016 Tue, 3:00–4:00PM, Eckhart 207

Peter McGrath, Brown University

New doubling constructions for minimal surfaces: I will discuss recent work (with Nikolaos Kapouleas) on constructions of new embedded minimal surfaces using singular perturbation 'gluing' methods. I will discuss at length doublings of the equatorial S^2 in S^3. In contrast to earlier work of Kapouleas, the catenoidal bridges may be placed on arbitrarily many parallel circles on the base S^2. This necessitates a more detailed understanding of the linearized problem and more exacting estimates on associated linearized doubling solutions.


Nov 22, 2016 Tue, 3:00–4:00PM, Eckhart 207 There will be two seminars on this date

Franco Vargas Pallete, UC Berkeley

Renormalized volume: The renormalized volume V_R is a finite quantity associated to certain hyperbolic 3-manifolds of infinite volume. In this talk I'll discuss its definition and some properties for acylindrical manifolds, namely local convexity and convergence under geometric limits. If time suffices I'll also discuss some applications and further examples.


Nov 22, 2016 Tue, 4:00–5:00PM, Eckhart 207 There will be two seminars on this date

Christos Mantoulidis, Stanford University

Fill-ins, extensions, scalar curvature, and quasilocal mass: There is a special relationship between the Jacobi and the ambient scalar curvature operator. I'll talk about an extremal bending technique that exploits this relationship. It lets us compute the Bartnik mass of apparent horizons and disprove a form of the Hoop conjecture due to G. Gibbons. Then, I'll talk about a derived "cut-and-fill" technique that simplifies 3-manifolds of nonnegative scalar curvature. It is used in studying a priori L^1 estimates for boundary mean curvature for compact initial data sets, and in generalizing Brown-York mass. Parts of this talk reflect work done jointly with R. Schoen/P. Miao.


Nov 29, 2016 Tue, 3:00–4:00PM, Eckhart 207

Hung Tran, UC Irvine

Index Characterization for Free Boundary Minimal Surfaces: A FBMS in the unit Euclidean ball is a critical point of the area functional among all surfaces with boundaries in the unit sphere, the boundary of the ball. The Morse index gives the number of distinct admissible deformations which decrease the area to second order. In this talk, we explain how to compute the index from data of two simpler problems. The first one is the corresponding problem with fixed boundary condition; and the second is associated with the Dirichlet-to-Neumann map for Jacobi fields. We also discuss applications to a conjecture about FBMS with index 4.



For further information on this seminar or this webpage, please contact Marco Guaraco at guaraco(at)math.uchicago.edu, Lucas Ambrozio at lambrozio(at)math.uchicago.edu or Rafael Montezuma at montezuma(at)math.uchicago.edu